Normal Line Load Research Articles

Investigation of the asperity point load mechanism for thermal elastohydrodynamic conditions

The rolling contact fatigue damage called pitting or spalling develops more frequently in surfaces with negative than positive slip. Since normal line loads do not cause any tensile surface stresses this investigation considers the effects of small point shaped asperities. Shear traction causes tensile stresses at the trailing edge of asperities entering the contact at negative slip. At positive slip the tensile stresses appear at the leading edge when the asperities exit the contact. It was found that the trailing edge of the asperity breaks through the lubrication film at contact entry. This causes negative slip to be more detrimental than positive slip. At negative slip the location of large frictional shear stresses and tension stresses from normal asperity contact coincide.

The effect of surface bending and surface stress on the transmission of a vertical line force in soft materials

When the surface of a soft substrate that carries a constant in-plane residual stress is indented by a concentrated line force, its profile near the applied load is found to have a kink, which results from a local balance of the surface stresses and the imposed force. Although the local bulk stresses in the substrate no longer have a net contribution to this force balance, they nevertheless grow according to a weak logarithmic singularity with respect to distance from the line load. Here we study how a normal line load is transmitted across a solid surface that can provide additional resistance due to bending deformation; we present an exact closed-form solution. Our analysis shows that the ability of the surface to resist bending completely regularizes the stress field — it is continuously differentiable everywhere. In particular, the stress state in the elastic substrate is hydrostatic right underneath the line load if the material is incompressible. Its maximum value is directly proportional to the applied normal load and inversely proportional to the elasto-bending length. It also depends on a dimensionless parameter which is the ratio of the elasto-capillary length to the elasto-bending length.

Lamb's problem on random mass density fields with fractal and Hurst effects.

This paper reports on a generalization of Lamb's problem to a linear elastic, infinite half-space with random fields (RFs) of mass density, subject to a normal line load. Both, uncorrelated and correlated (with fractal and Hurst characteristics) RFs without any weak noise restrictions, are proposed. Cellular automata (CA) is used to simulate the wave propagation. CA is a local computational method which, for rectangular discretization of spatial domain, is equivalent to applying the finite difference method to the governing equations of classical elasticity. We first evaluate the response of CA to an uncorrelated mass density field, more commonly known as white-noise, of varying coarseness as compared to CA's node density. We then evaluate the response of CA to multiscale mass density RFs of Cauchy and Dagum type; these fields are unique in that they are able to model and decouple the field's fractal dimension and Hurst parameter. We focus on stochastic imperfection sensitivity; we determine to what extent the fractal or the Hurst parameter is a significant factor in altering the solution to the planar stochastic Lamb's problem by evaluating the coefficient of variation of the response when compared with the coefficient of variation of the RF.

Three-Dimensional Vibration Analysis of Thick FGM Plate Strips Under Moving Line Loads

An exact three-dimensional analysis for steady-state dynamic response of an arbitrarily thick, isotropic, and functionally graded plate strip due to the action of a transverse distributed moving line load which is propagating parallel to the infinite simply supported edges of the plate at constant speed is presented based on the linear elasticity theory. The inhomogeneous plate is approximated by a laminate model, for which the solution is expected to gradually approach the exact one as the number of layers increases. The problem solution is derived by using Fourier transformation with respect to a moving reference frame in conjunction with the classical transfer matrix approach entailing the continuity of displacement and stress components at the interfaces of neighboring layers. The analytical results are illustrated with numerical examples in which a metal-ceramic (ZrO 2–Al) FGM plate strip of unit width is subjected to a half-sine normal line load of constant amplitude traveling along the strip at uniform speeds. Four types of FGM plate strips are configured, and the effects of load velocity, material compositional gradient, and plate thickness on the basic dynamic field quantities are evaluated and discussed. Also, the response curves for the FGM plates are compared with those of equivalent bilaminate plates containing comparable total volume fractions of constituent materials. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.

Application of the reciprocity theorem to determine line-load-generated surface waves on an inhomogeneous transversely isotropic half-space

Free and forced time-harmonic surface waves are investigated for a half-space of transversely isotropic elastic material, where the axis normal to the free surface (the z-axis) is the axis of symmetry. In addition, the elastic constants and the mass density depend on the distance from the free surface. A surface wave is considered as being composed of a carrier wave that propagates over the free surface as it would over a membrane, while it carries along a system of depth-dependent motions. For free surface waves the analytical details, i.e., the calculation of the wave speed of the free surface waves, are worked out for the case that the elastic constants and the mass density vary exponentially with depth. For forced motion by a time-harmonic line load, the far-field surface wave motion is determined by an application of the reciprocity theorem for elastodynamics. Numerical results for displacements and stresses are graphically displayed. Fourier superposition has been used to determine the transient surface wave response to a pulse-type normal line-load.

Debonding of a thermoelastic material from a rigid substrate at any constant speed: thermal relaxation effects

A linear isotropic thermoelastic half-space is debonded from a rigid insulated substrate at constant speed by moving shear and normal line loads. A dynamic steady state is examined, and an exact transform solution for the related problem of an insulated half-space subjected to a moving zone of specified surface displacements is obtained. Asymptotic forms are extracted that are valid near the zone edge and for high speeds, and which highlight thermal relaxation effects. They are used to derive analytical results for debonding at any constant speed. In particular, field variables on the debonded surface and the still-bonded interface are given for the sub-Rayleigh, super-Rayleigh/subsonic, lower and upper transonic, and supersonic speed ranges. The degenerate cases that arise at the three body wave speeds and at twice the rotational wave speed are also given. Calculations for the dynamic fracture energy rate and debonding zone temperature change at sub-Rayleigh speeds in 4340 steel indicate that thermal relaxation enhances energy rate, but mutes thermal response. The latter effect, however, itself decreases as the Rayleigh speed is approached.

Boussinesq–Flamant problem in gradient elasticity with surface energy

The strain gradient elasticity theory with surface energy is applied to Boussinesq–Flamant problem. The solution for the vertical displacements at the surface of half space due to the surface normal line load is presented. The theory includes both volumetric and surface energy terms. Boussinesq–Flamant problem in the strain gradient elasticity is solved by means of Fourier transform. The results obtained show that the vertical displacements of half space in the gradient elasticity are some different from that in the classical elasticity and the effects of the strain gradient parameters (material characteristic lengths) on the vertical displacements do exist.

Effects of Mixed-Mode and Crack Surface Convection in Rapid Crack Growth in Coupled Thermoelastic Solids

Two Green’s function problems for rapid two-dimensional steady-state crack growth governed by fully coupled (dynamic) linear thermoelasticity are analyzed. In Problem A, normal and in-plane shear line loads move on the insulated surfaces of a semi-infinite crack growing at a subcritical speed. Problem B involves only normal line loads, but crack surface convection is allowed. Problem A involves, therefore, mixed traction/displacement boundary conditions, while Problem B also exhibits mixed thermal boundary conditions. Robust asymptotic forms based on exact solutions for related problems reduce Problems A and B to coupled sets of integral equations. Both sets exhibit both Cauchy and Abel operators, but are solved exactly. The solutions show that Mode II loading couples the tangential crack face separation and discontinuity in crack-face temperature changes, while crack surface convection enhances thermal response, especially at large distances. [S0021-8936(00)03101-9]

Total Deformation, Plane-Strain Contact Analysis of Macroscopically Homogeneous, Compositionally Graded Materials With Constant Power-Law Strain Hardening

Plane-strain contact analysis is presented for compositionally graded materials with power-law strain hardening. The half-space, y≤0, is modeled as an incompressible, nonlinear elastic material. The effective stress, σe, and the effective total strain, εe, are related through a power-law model, σe=K0εeμ;0<μ≤min(1,(1+k)). The material property K0 changes with depth, |y|, as K0=A|y|k;A>0,0≤|k|<1. This material description attempts to capture some features of the plane-strain indentation of elastoplastic or steady-state creeping materials that show monotonically increasing or decreasing hardness with depth. The analysis starts with the solution for the normal line load (Flamant’s problem) and continues with the rigid, frictionless, flat-strip problem. Finally, the general solution of normal indentation of graded material by a convex, symmetric, rigid, and frictionless two-dimensional punch is given. Applications of the present results range from surface treatments of engineering structures, protective coatings for corrosion and fretting fatigue, settling of beam type foundations in the context of soil and rock mechanics, to bioengineering as well as structural applications such as contact of railroad tracks.

Transient wave propagation in a viscoelastic sandwich plate

This paper examines the propagation of coupled P and S waves in a three layer sandwich plate consisting of a viscoelastic core bounded by identical elastic outer layers. The waves are initiated by an impulsive normal line load applied to the upper surface of the plate. The response of the top surface is computed for various values of the creep and relaxation time constants of the core. The results indicate that, for some values of the time constants, the viscoelastic core provides effective damping of the transients, while for others the damping is virtually non-existent. In each case the dominant contribution to the far-field solution comes from the Rayleigh wave.

Static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads

The transfer matrix approach is used to solve the problem of static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads. The general problem is decoupled into two independent problems. The antiplane strain problem and the plane strain problem are considered in detail. Integral expressions for displacements and stresses at any point of the medium due to a normal line load and a shear line load, acting parallel to a symmetry axis, are obtained. In the case of a uniform half-space, closed form analytic expressions for displacements and stresses are derived. The procedure developed is quite easy and convenient for numerical computations.

Response of an elastically supported plate strip to a moving load

A theoretical analysis is presented of the steady state response of a plate strip constrained elastically along its edges against rotation and translation under the action of a moving transverse line load. Within the classical plate theory the solutions are obtained by using Fourier and Laplace transformation methods with respect to space variables. Numerical results are given for a plate strip with both edges identically constrained and a normal line load of constant intensity travelling along the plate strip with a constant speed. The first five speeds of the applied load for which a resonance effect occurs in the system are plotted as functions of the edge constraint parameters. The profiles of the displacement and the moment of the plate are also shown graphically for several values of the load speed and the edge constraint parameters.

Dynamic Whip of Elastically Restrained Plate Strips to Rapid Transit Loads

A plate strip of infinite length and constant width is cantilevered on a uniform elastic support along one edge and free along its opposite edge. A normal line load of constant intensity applied across its width travels along the strip at constant speed. Using plate theory, steady state solutions for the flexural waves are derived in terms of the generalized Fourier integral. Superposition is used to simulate responses to distributed transit loads. Results are applicable to the design of cantilevered guidance panels for air cushion vehicles and also in the design of the metal plates for the linear induction motors sometimes used to power these vehicles.

Near-field transient waves in anisotropic elastic plates for two and three dimensional problems

The formal solutions to a class of non-axisymmetric problems involving a transversely isotropic elastic plate are derived. The Cagniard-deHoop method of inversion is used to Obtain explicit solutions on the epicentral axis for certain materials, when a point shear load, with a “ramp” time-dependence, is applied to one face of the plate. Also treated is a problem involving plane-strain type motions, namely, the impulsive application of a normal line load to one face of an anisotropic plate, inversion for certain materials again being carried out on the epicentral axis. In both cases a numerical analysis of the solutions is presented.